An FEM approximation for a fourth-order variational inequality of second kind

10Wind-Turbine Aerodynamicsand FSICountries around the world are putting substantial effort into the development of wind energy technologies.The ambitious wind energy goals put pressure on the wind energy industry research and development to significantly enhance current wind generation capabilities in a short period of time and decrease the associated costs.This calls for transformative concepts and designs(e.g.,floating offshore wind turbines)that must be created and analyzed with high-precision methods and tools.These include complex-geometry,3D,time dependent,multi-physics predictive simulation methods and software that will play an increasingly important role as the demand for wind energy grows.Currently most wind-turbine aerodynamics and aeroelasticity simulations are performed using low-fidelity methods,such as the Blade Element Momentum(BEM)theory for the rotor aerodynamics employed in conjunction with simplified structural models of the wind-turbine blades and tower(see,e.g.,Jonkman and Buhl Jr.,2005;Jonkman et al.,2009).These methods are very fast to implement and execute.However,the cases involving unsteadyflow, turbulence,3D details of the wind-turbine blade and tower geometry,and other similarly-important features,are beyond their range of applicability.To obtain high-fidelity predictive simulation results for wind turbines,3D modeling is essential.However,simulation of wind turbines at full scale engenders a number of chal-lenges:theflow is fully turbulent,requiring highly accurate methods and increased grid resolution.The presence offluid boundary layers,where turbulence is created,complicates the situation further.Wind-turbine blades are long and slender structures,with complex distribu-tion of material properties,for which the numerical approach must have good approximation properties and avoid locking.Wind-turbine simulations involve moving and stationary com-ponents,and thefluid–structure coupling must be accurate,efficient and robust.These explain the current,modest nature of the state-of-the-art in wind-turbine simulations.FSI simulations at full scale are essential for accurate modeling of wind turbines.The motion and deformation of the wind-turbine blades depend on the wind speed and airflow, and the airflow patterns depend on the motion and deformation of the blades.In order to simulate the coupled problem,the equations governing the airflow and the blade motions and Computational Fluid–Structure Interaction:Methods and Applications,First Edition.Yuri Bazilevs,Kenji Takizawa and Tayfun E.Tezduyar.c 2013John Wiley&Sons,Ltd.Published2013by John Wiley&Sons,Ltd.316Computational Fluid–Structure Interaction:Methods and Applications deformations need to be solved simultaneously.Without that the modeling cannot be realistic: unsteady blade deformation affects aerodynamic efficiency and noise generation,and response to wind gusts.Flutter analysis of large blades operating in offshore environments is of great importance and cannot be accomplished without FSI.In recent years,several attempts were made to address the above mentioned challenges and to raise thefidelity and predictability levels of wind-turbine simulations.Stand alone aerody-namics simulations of wind-turbine configurations in3D were reported in Sørensen et al. (2002),Pape and Lecanu(2004),Zahle et al.(2009),Bazilevs et al.(2011b),Takizawa et al. (2011a,b)and Li and K.J.Paik(2012),while stand alone structural analyses of rotor blades of complex geometry and material composition,but under assumed wind-load conditions or wind-load conditions coming from separate aerodynamic computations were reported in Gut-tierez et al.(2003),Kong et al.(2005),Hansen et al.(2006),Jensen et al.(2006),Kiendl et al.(2010)and Bazilevs et al.(2012a).In the recent work of Bazilevs et al.(2011c)it was shown that coupled FSI modeling and simulation of wind turbines is important for accurately predicting their mechanical behavior at full scale.We feel that to address the above mentioned challenges one should employ a combination of numerical techniques,which are general,accurate,robust and efficient for the targeted class of problems.Such techniques are summarized in what follows and are described in greater detail in the body of this chapter.IGA is adopted as the geometry modeling and simulation framework for wind turbines in some of the examples presented in this chapter.We use IGA based on NURBS,which are more efficient than standardfinite elements for representing complex,smooth geometries, such as wind-turbine blades.The IGA was successfully employed for computation of turbulentflows(Bazilevs et al.,2007a,c,2010e;Akkerman et al.,2008;Hsu et al.,2010; Bazilevs and Akkerman,2010),nonlinear structures(Elguedj et al.,2008;Lipton et al.,2010; Benson et al.,2010a,b;Kiendl et al.,2009,2010),and FSI(Zhang et al.,2007;Bazilevs et al.,2006a,2008;Isaksen et al.,2008),and,in most cases,gave a clear advantage over standard low-orderfinite elements in terms of solution accuracy per degree-of-freedom.This is in part attributable to the higher-order smoothness of the basis functions employed.Flows about rotating components are naturally handled in an isogeometric framework because all conic sections,and in particular,circular and cylindrical shapes,are represented exactly (Bazilevs and Hughes,2008).The blade structure is governed by the isogeometric rotation-free shell formulation with the aid of the bending-strip method(Kiendl et al.,2010).The method is appropriate for thin-shell structures comprised of multiple C1-or higher-order continuous surface patches that are joined or merged with continuity no greater than C0.The Kirchhoff–Love shell theory that relies on higher-order continuity of the basis functions is employed in the patch interior as in Kiendl et al.(2009).Although NURBS-based IGA is employed in this work,other discretizations such as T-splines(Bazilevs et al.,2010a;Dörfel et al.,2010)or subdivision surfaces(Cirak et al.,2000,2002;Cirak and Ortiz,2001),are perfectly suited for the proposed structural modeling method.In addition,an isogeometric representation of the analysis-suitable geometry may be used to construct tetrahedral and hexahedral meshes for computations using the FEM.In this chapter,we use such tetrahedral meshes for wind-turbine rotor computation using the ALE-VMS(see Section4.6.1)and DSD/SST-VMST(ST-VMS)(see Section4.6.3)methods.In application of the DSD/SST formulation toflows with moving mechanical components,theWind-Turbine Aerodynamics and FSI317 Shear–Slip Mesh Update Method(SSMUM)(Tezduyar et al.,1996;Behr and Tezduyar,1999, 2001)has been very instrumental.The SSMUM wasfirst introduced for computation offlow around two high-speed trains passing each other in a tunnel(see Tezduyar et al.,1996).The challenge was to accurately and efficiently update the meshes used in computations based on the DSD/SST formulation and involving two objects in fast,linear relative motion.The idea behind the SSMUM was to restrict the mesh moving and remeshing to a thin layer of elements between the objects in relative motion.The mesh update at each time step can be accomplished by a“shear”deformation of the elements in this layer,followed by a“slip”in node connectiv-ities.The slip in the node connectivities,to an extent,un-does the deformation of the elements and results in elements with better shapes than those that were shear-deformed.Because the remeshing consists of simply re-defining the node connectivities,both the projection errors and the mesh generation cost are minimized.A few years after the high-speed train computa-tions,the SSMUM was implemented for objects in fast,rotational relative motion and applied to computation offlow past a rotating propeller(Behr and Tezduyar,1999)andflow around a helicopter with its rotor in motion(Behr and Tezduyar,2001).A number of special techniques for wind-turbine aerodynamics and FSI simulation were developed recently.In Bazilevs et al.(2011b),a technique for describing the wind-turbine rotor geometry was developed based on NURBS and applied to the aerodynamics simulation of a5MW wind-turbine rotor with its design specified in Jonkman et al.(2009).Follow-up computational studies of this rotor were presented in Takizawa et al.(2011a,b)and Hsu et al.(2011b).In Hsu et al.(2011a),the aerodynamics simulations of the National Renew-able Energy Lab(NREL)Phase VI two-bladed rotor(see Hand et al.,2001)were performed to validate the ALE-VMS computations against an extensive set of experimental data avail-able for this test case.The structural mechanics formulation for wind-turbine blades,which is based on the Kirchhoff–Love thin shell theory and the bending strip method(see Kiendl et al.,2009,2010),was developed and applied in Bazilevs et al.(2011c)to fully-coupled FSI simulation of a5MW wind-turbine rotor.We believe this was thefirst fully-coupled FSI simu-lation of a full-scale wind-turbine rotor.A special mesh moving procedure for FSI simulation of wind-turbine rotors was also proposed in Bazilevs et al.(2011c),where only the deflection part of the mesh motion is handled using the elasticity-based mesh moving method(Tezduyar et al.,1992b,1993),while the rotational part is handled exactly.A method and algorithm for pre-bending of wind-turbine blades to avoid the blade striking the tower during operation in high winds were recently proposed in Bazilevs et al.(2012a).In this chapter we provide a description of these techniques.10.1Aerodynamics Simulations of a5MW Wind-Turbine RotorIn this section we begin with a careful definition of the5MW wind-turbine rotor geometry. We then present the NURBS-based and FEM-based simulations of the wind-turbine rotor.In this section,we only present pure aerodynamic simulations.Structural and FSI modeling and simulations will be presented in the later sections.10.1.15MW Wind-Turbine Rotor Geometry DefinitionAs afirst step we construct a template for the structural model of the rotor.Here,the structural model is limited to a surface(shell)representation of the wind-turbine blade,the hub,and their318Computational Fluid–Structure Interaction:Methods and Applications Table10.1Wind-turbine rotor geometry definitionRNodes(m)AeroTwst(◦)Chord(m)AeroCent(-)AeroOrig(-)Airfoil2.00000.0003.5420.25000.50Cylinder2.86670.0003.5420.25000.50Cylinder5.60000.000 3.8540.22180.44Cylinder8.33330.000 4.1670.18830.38Cylinder11.750013.308 4.5570.14650.30DU4015.850011.480 4.6520.12500.25DU3519.950010.162 4.4580.12500.25DU3524.05009.011 4.2490.12500.25DU3028.15007.795 4.0070.12500.25DU2532.2500 6.544 3.7480.12500.25DU2536.3500 5.361 3.5020.12500.25DU2140.4500 4.188 3.2560.12500.25DU2144.5500 3.125 3.0100.12500.25NACA6448.6500 2.310 2.7640.12500.25NACA6452.7500 1.526 2.5180.12500.25NACA6456.16670.863 2.3130.12500.25NACA6458.90000.370 2.0860.12500.25NACA6461.63330.106 1.4190.12500.25NACA6462.90000.0000.7000.12500.25NACA64 attachment zone.The blade surface is assumed to be composed of a collection of airfoil shapes that are lofted in the blade-axis direction.The geometry of the rotor blade is based on the NREL5MW offshore baseline wind turbine described in Jonkman et al.(2009).The blade geometry data taken from the reference is summarized in Table10.1.A61m blade is attached to a hub with radius of2m,which gives the total rotor radius of63m.The blade is composed of several airfoil types provided in the rightmost column of the table.Thefirst portion of the blade is a perfect cylinder.Further away from the root the cylinder is smoothly blended into a series of DU(Delft University)airfoils. At the44.55m location away from the root the NACA64profile is used to define the blade all the way to the tip(see Figure10.1).The remaining parameters from Table10.1are defined in Figure10.1:“RNodes”is the distance from the rotor center to the airfoil cross-section in the blade axis direction.“AeroTwst”is the twist angle for a given cross-section.The blades are twisted to enhance the aerodynamic performance.“Chord”is the chord length of the airfoil.“AeroOrig”is the location of the aerodynamic center.For most of the blade airfoil cross-sections,the aerodynamic center is taken at25%of the chord length from the leading edge. To accommodate the cylindrical shape at the root,the aerodynamic center is gradually moved to50%of the chord length.This is not reported in Jonkman et al.(2009),but mentioned in Kooijman et al.(2003).Remark10.1There is some redundancy in the parameters given in Table10.1.The vari-able“AeroCent”is used as an input to FAST(Jonkman and Buhl Jr.,2005),which is the aerodynamics modeling software that is typically used for wind-turbine rotor computations. FAST assumes that the blade-pitch axis passes through each airfoil section at25%chord length,and defines AeroCent−0.25to be the fractional distance to the aerodynamic centerWind-Turbine Aerodynamics and FSI319Figure10.1Illustration of quantities from Table10.1from the blade-pitch axis along the chordline,positive toward the trailing edge.Therefore, AeroOrig+(0.25−AeroCent)gives the location of where the blade-pitch axis passes through each airfoil cross-section.Although for our purposes this added complexity is unnecessary, the same naming system is used for backward compatibility with the referenced reports.For each blade cross-section,we use quadratic NURBS to represent the2D airfoil shape. The weights of the NURBS functions are set to unity.The weights are adjusted near the root to represent the circular cross-sections of the blade exactly.The cross-sections are lofted in the blade axis direction,also using quadratic NURBS and unity weights.This geometry modeling procedure produces a smooth rotor-blade surface using a relatively small number of input parameters,which is an advantage of the isogeometric representation.Figure10.2 shows a top view of the blade in which the twisting of the cross-sections is evident.GivenFigure10.2Top view of a subset of the airfoil cross-sections illustrating blade twisting320Computational Fluid–Structure Interaction:Methods and Applications the rotor-blade surface description,the fluid-domain volume is constructed next.The blade surface is split into four patches of similar sizes,which we call the blade surface patches.The splitting is done at the leading and trailing edges,as well as half-way in between on both sides of the blade.The fluid domain near the blade is generated for each one of the blade surface patches.As a final step,the fluid-domain patches are merged such that the outer boundary of the domain is a perfect cylinder.For each of the blade surface patches,we create a 60◦pie-shaped domain using a minimum required number of control points.The control points at the bottom of the patch are moved to accommodate the shape of the rotor hub.As a next step,we perform knot insertion and move the new control points such that their locations coincide with those of the blade surface patch.This generates an a priori conforming discretization between the volume fluid domain and the surface of the structural model,suitable for FSI analysis.Finally,the fluid domain is refined in all parametric directions for analysis.Figure 10.3a shows the rotor surface mesh and one of the fluid-mesh subdomains adjacent to it.The remaining fluid subdomains are generated in the samemanner.(a)63 m40.95 m35.79 m 97.85 m(b)Figure 10.3(a)V olume NURBS mesh of the computational domain.(b)A planar cut to illustrate mesh grading toward the rotor bladeThe resultant fluid NURBS mesh may be embedded into a larger domain for the purposes of simulation.In this work we take this larger domain to also be a cylinder.For computational e fficiency,only one-third of the domain is modeled.The fluid volume mesh,corresponding to one-third of the fluid domain,consists of 1449000quadratic NURBS elements (and a similar number of control points).The fluid mesh cross-section that also shows the details of mesh refinement in the boundary layer is shown in Figure 10.3b.To carry out the simulations,rotational-periodicity conditions (see Sections 9.2and 9.2.3)are imposed.Denoting by u h l andu hl=R(2/3π)u h r,(10.2) where R(2/3π)is the rotation matrix evaluated atα=2/3π.That is,while the pressure degrees-of-freedom take on the same values,thefluid velocity degrees-of-freedom are related through a linear transformation corresponding to a rotation by2/3πradians.Note that the transformation matrix is independent of the current domain position.Rotational-periodicity conditions are implemented through standard master-slave relationships.We note that rotational-periodicity conditions were employed earlier in Takizawa et al.(2011d,f,2010c) for parachute simulations.We compute the aerodynamics of the wind-turbine rotor with prescribed speed using a rotating mesh.The wind speed is uniform at9m/s and the rotor speed is1.08rad/s,giving a tip speed ratio of7.55(see Spera,1994for wind-turbine terminology).We use air propertiesat standard sea-level conditions.The Reynolds number(based on the cord length at34R andthe relative velocity there)is approximately12million.At the inflow boundary the velocity is set to the wind velocity,at the outflow boundary the stress vector is set to zero,and at the radial boundary the radial component of the velocity is set to zero.We start from aflowfield where the velocity is equal to the inflow velocity everywhere in the domain except on the rotor surface,where the velocity matches the rotor velocity.We carry out the computations at a constant time-step size of4.67×10−4s.Both NURBS and tetrahedral FEM simulations of this setup are performed.The chosen wind velocity and rotor speed correspond to one of the cases given in Jonkman et al.(2009),where the aerodynamics simulations were performed using FAST(Jonkman and Buhl Jr.,2005).We note that FAST is based on lookup tables for airfoil cross-sections,which give planar,steady-state lift and drag data for a given wind speed and angle of attack.The effects of trailing-edge turbulence,hub,and tip are incorporated through empirical models. It was reported in Jonkman et al.(2009)that at these wind conditions and rotor speed,no blade pitching takes place and the rotor develops a favorable aerodynamic torque(i.e.,torque322Computational Fluid–Structure Interaction:Methods and Applications in the direction of the rotation)of2500kN m.Although this value is used for comparison with our simulations,the exact match is not expected,as our computational modeling is very different than the one in Jonkman et al.(2009).Nevertheless,we feel that this value of the aerodynamic torque is close to what is expected in reality,given the vast experience of NREL with wind-turbine rotor simulations employing FAST.10.1.2ALE-VMS Simulations Using NURBS-based IGAThe computation is carried out on240cores on Ranger,a Sun Constellation Linux Cluster at the Texas Advanced Computing Center(TACC)with62976processing cores.Near the blade surface,the size of thefirst element in the wall-normal direction is about2cm.The GMRES search technique Saad and Schultz(1986)is used with a block-diagonal preconditioner.Each nodal block consists of a3×3and1×1matrices,corresponding to the discrete momentum and continuity equations,respectively.The number of nonlinear iterations per time step is 4and the number of GMRES iterations is200for thefirst nonlinear iteration,300for the second,and400for the third and fourth.Figure10.5shows the air speed at t=0.8s atFigure10.5Air speed at t=0.8s1m behind the rotor plane.Note thefine-grained turbulent features at the trailing edge of the blade,which require enhanced mesh resolution for accurate representation.Thefluid-traction vectors projected to the plane of rotation are shown in Figure10.6.The traction vectors point in the direction of rotation and grow in magnitude toward the blade tip,creating favorable aerodynamic torque.However,at the blade tip the traction vectors rapidly decay to zero and even change sign,which introduces a small amount of inefficiency.The time history of the aerodynamic torque is shown in Figure10.7,where the steady-state result from Jonkman et al.(2009)is also shown for reference.Thefigure shows that in less than0.8s the torque settles at a statistically-stationary value of2670kN m,which is within6.4%of the reference. Given the significant differences in the computational modeling approaches,the two valuesWind-Turbine Aerodynamics and FSI323Figure10.6Fluid traction vectors at t=0.8s viewed from the back of the blade.Thefluid-traction vectors,colored by magnitude,are projected to the rotor plane and illustrate the mechanism by which the favorable aerodynamic torque is createdFigure10.7Time history of the aerodynamic torque.Statistically-stationary torque is attained in less than0.8s.The reference steady-state result from NREL is also shown for comparisonare remarkably close.This result is encouraging in that3D time-dependent simulation with a manageable number of degrees-of-freedom and without any empiricism is able to predict important quantities of interest for wind-turbine rotors simulated at full scale.This result also gives us confidence that our procedures are accurate and may be applied to simulations cases where3D,time-dependent modeling is indispensable(e.g.,simulation of wind gusts or blade pitching).324Computational Fluid–Structure Interaction:Methods and Applications Given the aerodynamic torque and the rotor speed,the power extracted from the wind with these wind conditions(based on our aerodynamic torque T f)isP=T f˙θ≈2.88MW.(10.3) According to the Betz law(see,e.g.,Hau,2006),the maximum power that a horizontal-axis wind turbine is able to extract from the wind isP max=1627ρA u in 32≈3.23MW,(10.4)where A=πR2is the cross-sectional area swept by the rotor,and u in is the inflow speed. From this we conclude that the wind-turbine aerodynamic efficiency at the simulated wind conditions isPP max≈89%,(10.5)which is quite high even for modern wind-turbine designs.The blade is segmented into18 spanwise“patches”to investigate how the aerodynamic torque distribution varies along the blade span.The patch-wise torque distribution is shown in Figure10.8.The torque is nearly zero in the cylindrical section of the blade.A favorable aerodynamic torque is created on Patch 4and its magnitude continues to increase until Patch15.The torque magnitude decreases rapidly after Patch15,however,the torque remains favorable all the way to the last patch.The importance of3D modeling and simulation is further illustrated in Figure10.9,where the axial component of theflow velocity is displayed at a blade cross-section located at56m above the rotor center.The magnitude of the axial velocity component exceeds15m/s in the123456789101112131415161718Figure10.8Patches along the blade(top)and the aerodynamic torque contribution from each patch (bottom)at t=0.8sFigure10.9Axialflow velocity over the blade cross-section at56m at t=0.8s.The level of axial flow in the boundary layer is significant,which illustrates the importance of3D modelingboundary layer,showing that3D effects are important,especially in the regions of the blade with the largest contribution to the aerodynamic torque.10.1.3Computations with the DSD/SST Formulation Using Finite ElementsWe describe from Takizawa et al.(2011a,b)the computations with the DSD/SST formulation and linearfinite elements.To generate the triangular mesh on the rotor surface,we started with a quadrilateral surface mesh generated by interpolating the NURBS geometry of the rotor at each knot intersection.We subdivided each quadrilateral element into triangles and then made minor modifications to improve the mesh quality near the hub.We use three different meshes:Mesh-2,Mesh-3and Mesh-4,with the surface mesh refined along the blade2,3and 4times,respectively,compared to thefinite element mesh used in Bazilevs et al.(2011b). The number of nodes and elements for each blade surface mesh is shown in Table10.2,and Figure10.10shows the surface mesh for Mesh-4.For computational efficiency,rotational-periodicity(Takizawa et al.,2011d,f)is utilized so that the domain includes only one of three blades,as shown in Figure10.11.The inflow,outflow and radial boundaries lie0.5R,2R,andTable10.2Summary of the meshes,where nn and ne are thenumber of nodes and elementsSurface V olumenn ne nn neMesh-2574811452155494898640Mesh-37552150602058551195452Mesh-49268184922533401475175Figure10.11Rotationally-periodic domain with wind-turbine blade shown in blueFigure10.12Cut plane of thefluid volume mesh along rotor axis(Mesh-4)1.43R from the hub center,respectively.This can be more easily seen in Figure10.12,where the inflow,outflow,and radial boundaries are the left,right and top edges,respectively,of the cut plane along the rotation axis.Each periodic boundary contains1430nodes and2697 triangles.Near the rotor surface,we have22layers of refined mesh withfirst-layer thickness ofFigure10.13Boundary layer mesh at34R1cm and a progression factor of1.1.The boundary layer mesh at34R is shown in Figure10.13.The number of nodes and elements for each volume mesh is shown in Table10.2.We compute the problem with the DSD/SST-SUPS and the conservative form of DSD/SST-VMST.The SUPS version is used without the LSIC stabilization(i.e.,νLSIC=0),while for the VMST versionνLSIC is defined according to Equation(4.126),and is referred to as“TGI.”In solving the linear equation systems involved at every nonlinear iteration,the GMRES search technique(Saad and Schultz,1986)is used with a diagonal preconditioner.The compu-tation is carried out in a parallel computing environment.The mesh is partitioned to enhance the parallel efficiency of the computations.Mesh partitioning is based on the METIS algo-rithm(Karypis and Kumar,1998).The time-step size is4.67×10−4s.The number of nonlinear iterations per time step is3with30,60,and500GMRES iterations for thefirst,second,and third nonlinear iterations,respectively.Prior to the computations reported here,we performed a series of brief computations with the DSD/SST-SUPS technique,starting from a lower Reynolds number and gradually reach-ing the actual Reynolds number.This solution is used as the initial condition also for the computations with the DSD/SST-VMST technique.The purpose is to generate a divergence-free and reasonableflowfield at this Reynolds number.We note that it was especially difficult with the VMST option to start from nonphysical conditions,such as setting all nodes except those on the blade to the inflow velocity.Figures10.14–10.16show the time history of the aerodynamic torque and the torque con-tribution from each patch for a single blade at t=1.0s.The patches are defined as shown in Figure10.8.Figure10.17shows the pressure coefficient at t=1.0s for Patch16(at0.90R), which is a representative section of the blade.For most of the patches,the angle of attack and Reynolds number do not vary much from one patch to another.For example,the angle of attack and Reynolds number are7.4◦and9.9×106at0.65R for Patch12(at0.65R)and7.6◦and9.6×106for Patch16(at0.90R).Mesh refinement studies for both the SUPS and VMST versions indicate good convergence in the quantities of interest such as the aerodynamic torque and pressure coefficient.The VMST version on thefinest mesh gives more or less the same value of the aerodynamic torque as the ALE-VMS simulation using NURBS,which is taken as a reference solution for this study.The results from the SUPS version are also very good,however the torque is slightly underpredicted with respect to the VMST and NURBS-based ALE-VMS simulations.。

T.TanakaDepartment of Electronics and Information EngineeringTokyo Metropolitan University1-1,Minami-Osawa,Hachioji,Tokyo192-0397JapanAbstractI present a theory of meanfield approximation based on information ge-ometry.This theory includes in a consistent way the naive meanfieldapproximation,as well as the TAP approach and the linear response the-orem in statistical physics,giving clear information-theoretic interpreta-tions to them.1INTRODUCTIONMany problems of neural networks,such as learning and pattern recognition,can be cast into a framework of statistical estimation problem.How difficult it is to solve a particular problem depends on a statistical model one employs in solving the problem.For Boltzmann machines[1]for example,it is computationally very hard to evaluate expectations of state variables from the model parameters.Meanfield approximation[2],which is originated in statistical physics,has been frequently used in practical situations in order to circumvent this difficulty.In the context of statistical physics several advanced theories have been known,such as the TAP approach[3],linear response theorem[4],and so on.For neural networks,application of meanfield approxi-mation has been mostly confined to that of the so-called naive meanfield approximation, but there are also attempts to utilize those advanced theories[5,6,7,8].In this paper I present an information-theoretic formulation of meanfield approximation.It is based on information geometry[9],which has been successfully applied to several prob-lems in neural networks[10].This formulation includes the naive meanfield approximation as well as the advanced theories in a consistent way.I give the formulation for Boltzmann machines,but its extension to wider classes of statistical models is possible,as described elsewhere[11].2BOLTZMANN MACHINESA Boltzmann machine is a statistical model with binary random variables,.The vector is called the state of the Boltzmann machine.The state is also a random variable,and its probability law is given by the Boltzmann-Gibbs distribution(1) where is the“energy”defined by(2) with and the parameters,and is determined by the normalization condition and is called the Helmholtz free energy of.The notation means that the summation should be taken over all distinct pairs.Let and,where means the expectation with respect to .The following problem is essential for Boltzmann machines:Problem1Evaluate the expectations and from the parameters and of the Boltzmann machine.3INFORMATION GEOMETRY3.1ORTHOGONAL DUAL FOLIATIONSA whole set of the Boltzmann-Gibbs distribution(1)realizable by a Boltzmann machine is regarded as an exponential family.Let us use shorthand notations,,,to represent distinct pairs of indices,such as.The parameters and constitute a coordinate system of,called the canonical parameters of.The expectations and constitute another coordinate system of,called the expectation parameters of.Let be a subset of on which are all equal to zero.I call the factorizable submodel of since can be factorized with respect to.On the problem is easy:Since are all zero,are statistically independent of others,and therefore and hold.Meanfield approximation systematically reduces the problem onto the factorizable sub-model.For this reduction,I introduce dual foliations and onto.The foliation ,,is parametrized by and each leaf is defined as(3) The leaf is the same as,the factorizable submodel.Each leaf is again an exponential family with and the canonical and the expectation parameters,respec-tively.A pair of dual potentials is defined on each leaf,one is the Helmholtz free energy and another is its Legendre transform,or the Gibbs free energy,(4) and the parameters of are given by(5) where and.Another foliation, ,is parametrized by and each leaf is defined as(6)Each leaf is not an exponential family,but again a pair of dual potentials and is defined on each leaf,the former is given by(7) and the latter by its Legendre transform as(8) and the parameters of are given by(9) where and.These two foliations form the orthogonal dual foliations,since the leaves and are orthogonal at their intersecting point.I introduce still another coordinate system on,called the mixed coordinate system,on the basis of the orthogonal dual foliations.It uses a pair of the expectation and the canonical parameters to specify a single element.The part specifies the leaf on which resides,and the part specifies the leaf.3.2REFORMULATION OF PROBLEMAssume that a target Boltzmann machine is given by specifying its parameters and .Problem1is restated as follows:evaluate its expectations and from those parameters.To evaluate meanfield approximation translates the problem into the following one:Problem2Let be a leaf on which resides.Find which is the closest to.Atfirst sight this problem is trivial,since one immediatelyfinds the solution.How-ever,solving this problem with respect to is nontrivial,and it is the key to understand-ing of meanfield approximation including advanced theories.Let us measure the proximity of to by the Kullback divergence3.3PLEFKA EXPANSIONThe problem is easy if.In this case is given explicitly as a function of asFor the second-order,(18)and(19)For the third-order,(20)and for,,for three distinct indices,,and,(21)For other combinations of,,and,(22) 4MEAN FIELD APPROXIMATION4.1MEAN FIELD EQUATIONTruncating the Plefka expansion(14)up to-th order term gives-th order approximations, and.The Weiss free energy,which is used in the naive meanfield approximation,is given by.The TAP approach picks up all relevant terms of the Plefka expansion[12],and for the SK model it gives the second-order approximation .The stationary condition gives the so-called meanfield equation,from which a solution of the approximate minimization problem is to be determined.For it takes the following familiar form,(23) and for it includes the so-called Onsager reaction term.(24) Note that all of these are expressed as functions of.Geometrically,the meanfield equation approximately represents the“surface”in terms of the mixed coordinate system of,since for the exact Gibbs free energy ,the stationary condition gives.Accordingly,the ap-proximate relation,forfixed,represents the-th order approximate expression of the leaf in the canonical coordinate system.Thefit of this expression to the true leaf around the point becomes better as the order of approximation gets higher,as seen in Fig.1.Such a behavior is well expected,since the Plefka expansion is essentially a Taylor expansion.4.2LINEAR RESPONSEFor estimating one can utilize the linear response theorem.In information geomet-rical framework it is represented as a trivial identity relation for the Fisher information on the leaf.The Fisher information matrix,or the Riemannian metric tensor,on the leaf,and its inverse are given by(25)η12η1=η200.250.50.75100.51A (m )F 00th order 1st order 2nd order 3rd order 4th order η12η1=η20.10.20.30.40.4990.50.501F 0A (m )Figure 1:Approximate expressions ofby mean field approximations of several or-ders for 2-unit Boltzmann machine,with(left),and their magnifiedview (right).Figure 2:theory.and(26)respectively.In the framework here,the linear response theorem states the trivial fact that those are the inverse of the other.In mean field approximation,one substitutes an approxi-mation in place of in eq.(26)to get an approximate inverse of the metric .The derivatives in eq.(26)can be analytically calculated,and therefore can be numer-ically evaluated by substituting to it a solution of the mean field equation.Equating its inverse togives an estimate of by using eq.(25).So far,Problem 1has been solved within the framework of mean field approximation,with and obtained by the mean field equation and the linear response theorem,respectively.5DISCUSSIONFollowing the framework presented so far,one can in principle construct algorithms of mean field approximation of desired orders.The first-order algorithm with linear response has been first proposed and examined by Kappen and Rodr´ıguez[7,8].Tanaka[13]has formulated second-and third-order algorithms and explored them by computer simulations.It is also possible to extend the present formulation so that it can be applicable to higher-order Boltzmann machines.Tanaka[14]discusses an extension of the present formulation to third-order Boltzmann machines:It is possible to extend linear response theorem to higher-orders,and it allows us to treat higher-order correlations within the framework of mean field approximation.The common understanding about the“naive”meanfield approximation is that it minimizes Kullback divergence with respect to for a given.It can be shown that this view is consistent with the theory presented in this paper.Assume thatand,and let be a distribution corresponding the intersecting point of the leaves and.Because of the orthogonality of the two foliations and the following“Pythagorean law[9]”holds(Fig.2).(27) Intuitively,measures the squared distance between and,and is a second-order quantity in.It should be ignored in thefirst-order approximation,and thus holds.Under this approximation minimization of the former with respect to is equivalent to that of the latter with respect to,which establishes the re-lation between the“naive”approximation and the present theory.It can also be checked directly that thefirst-order approximation of exactly gives,the Weiss free energy.The present theory provides an alternative view about the validity of meanfield approx-imation:As opposed to a common“belief”that meanfield approximation is a good one when is sufficiently large,one can state from the present formulation that it is so when-ever higher-order contribution of the Plefka expansion vanishes,regardless of whetheris large or not.This provides a theoretical basis for the observation that meanfield approx-imation often works well for small networks.The author would like to thank the Telecommunications Advancement Foundation forfi-nancial support.References[1]Ackley,D.H.,Hinton,G.E.,and Sejnowski,T.J.(1985)A learning algorithm for Boltzmannmachines.Cognitive Science9:147–169.[2]Peterson,C.,and Anderson,J.R.(1987)A meanfield theory learning algorithm for neuralplex Systems1:995–1019.[3]Thouless,D.J.,Anderson,P.W.,and Palmer,R.G.(1977)Solution of‘Solvable model of aspin glass’.Phil.Mag.35(3):593–601.[4]Parisi,G.(1988)Statistical Field Theory.Addison-Wesley.[5]Galland,C.C.(1993)The limitations of deterministic Boltzmann machine work4(3):355–379.[6]Hofmann,T.and Buhmann,J.M.(1997)Pairwise data clustering by deterministic annealing.IEEE Trans.Patt.Anal.&Machine Intell.19(1):1–14;Errata,ibid.19(2):197(1997).[7]Kappen,H.J.and Rodr´ıguez,F.B.(1998)Efficient learning in Boltzmann machines usinglinear response theory.Neural Computation.10(5):1137–1156.[8]Kappen,H.J.and Rodr´ıguez,F.B.(1998)Boltzmann machine learning using meanfield theoryand linear response correction.In M.I.Jordan,M.J.Kearns,and S.A.Solla(Eds.),Advances in Neural Information Processing Systems10,pp.280–286.The MIT Press.[9]Amari,S.-I.(1985)Differential-Geometrical Method in Statistics.Lecture Notes in Statistics28,Springer-Verlag.[10]Amari,S.-I.,Kurata,K.,and Nagaoka,H.(1992)Information geometry of Boltzmann ma-chines.IEEE Trans.Neural Networks3(2):260–271.[11]Tanaka,rmation geometry of meanfield approximation.preprint.[12]Plefka,P.(1982)Convergence condition of the TAP equation for the infinite-ranged Ising spinglass model.J.Phys.A:Math.Gen.15(6):1971–1978.[13]Tanaka,T.(1998)Meanfield theory of Boltzmann machine learning.Phys.Rev.E.58(2):2302–2310.[14]Tanaka,T.(1998)Estimation of third-order correlations within meanfield approximation.In S.Usui and T.Omori(Eds.),Proc.Fifth International Conference on Neural Information Process-ing,vol.1,pp.554–557.。

Microscopic origin of the second law of thermodynamicsYou-gang FengDepartment of Basic Science, College of Science , Guizhou University Guiyang, Cai Jia Guan, 550003, ChinaAbstractWe proved when random-variable fluctuations obey the central limit theorem the equality of the uncertainty relation corresponds to the thermodynamic equilibrium state. The inequality corresponds to the thermodynamic non-equilibrium state. The uncertainty relation is a quantum-mechanics expression of the second law of thermodynamics originated in wave-particle duality. Formulas of mean square-deviations changes adjusted by random fluctuations under the minimal uncertainty relation are obtained. Finally, an assumption is made which is waiting for examination. We except phase transitions in our discussion.Keywords: second law of thermodynamics, uncertainty relationPACS: 05.30.-d, 05.70.-aIt is well-known the uncertainty relation and Sch Ödinger’s equation are two foundations of quantum mechanics []1, they are in dependent of each other because one cannot be derived from the other. The relation revealed a restrictive relation of quantum fluctuations between positions and momentums. A perplexed problem is which thermodynamic state the equality of the uncertainty relation corresponds to, which thermodynamic state its inequality does to. Obviously, the equation and the relation themselves cannot solve it. The same problem is also met in the quantum statistical mechanics. The probability-density operator of a mixed ensemble is denoted by []2∑〉〈=i i i i ||ψψρρ, ∑=ii 1ρ, (1)the wave function 〉i ψ| of Eq.(1) is given by the Sch Ödinger’s equation, i ρ is the probability of subsystem of the ensemble in the state 〉i ψ|, i takes all possiblevalues, and each subsystem is independent of another and there is not any coherence between different states. The operator ρ does not relate to the uncertainty relation directly, namely, it cannot tell us what will actually happen to the relation in the equilibrium state. In the quantum statistical mechanics the quantum fluctuations cannot be neglected, which means there must be other probability-density operator F concerned in the relation, which also describes the equilibrium state, F and ρ will become two foundations of the quantum statistical mechanics as if the equation and the relation were in the quantum mechanics.The second law of thermodynamics has been regarded as a macroscopic law since it was found ，its microscopic origin and corresponding principle in the quantum mechanics have not been obtained. Some authors tried study this subject from the dynamic point of view ， but have never achieved a certain and satisfactory conclusion []3. In fact ，the process from the non-equilibrium state (time is ordered) to the equilibrium state (time is disordered) is a mutation ，which cannot be solved by means of dynamic equations. Expanding S in powers of random variables about the equilibrium-state entropy 0S ， Einstein []2 obtained Gaussian distribution of the fluctuations; Prigogine derived the minimal entropy-production principle with the same method []4. Since S is non-equilibrium–state entropy both of theories merely suit to the non-equilibrium-state fluctuations. The theories pointed out the transitions of entropies are linked to the fluctuations. We think the second law of thermodynamics maybe results from the fluctuations. According to Landau’sexplanation Einstein’s theory cannot be applied to the quantum statistical mechanics for it to neglect quantum effects []2, and the variables’ deviation only corresponds to the non-equilibrium state in his theory. It is proved the uncertainty relation is topologically invariant []5, which reminds us that the invariance of the minimal uncertainty relation maybe relates to the equilibrium-state entropy. Thus, we will start off with the uncertainty relation to discuss the fluctuations, except phase transitions.For one-dimension, mean square-deviations of the state |〉i ψ are〉〉〈−〈=∆22)()(i i x x x , 〉〉〈−〈=∆22)()(i i x x x p p p (2)〉〈i x and 〉〈i x p are its average values of positions and momentum. The statisticalfluctuations of random variables about the average values have the following common characters: On the one hand, any quantum system, no matter what its Hamiltonian operator is, obeys the uncertainty relation,222)4()()(πh p x i x i ≥∆∆ (3)(i x ∆)2and (i x p ∆)2 are given by Eq.(2), the equality of Eq.(3) is named theminimal uncertainty relation. On the other hand, the wave function describes the statistical behavior of the large number of particles, although some specific particles’ behavior violates the properties of the wave function. Let a subsystem be in the state |〉i ψ which particles’ number be i N , x and x p be the position and momentum of the state. Since the wave-function equation is distinct from a particle’s dynamic equation to describe the particle’s moving orbits and the wave function |〉i ψ only has statistical meaning, x and x p are the variables of the function |〉i ψ and they are not a specific particle’s position and momentum. In such a subsystem each actual particle has itself specific position and momentum caused by various reasons: collisions, transitions among energies levels, interactions of electrons with atomic nuclei, interactions between electrons, etc. Each reason results in a specific fluctuation of position or of momentum. How can the fluctuations )(〉〈−i x x and )(〉〈−i x x p p represent these fluctuations? Because the number of the particles isvery large all of these fluctuations can take place at the same time for a particle, and one specific fluctuation cannot be distinguished from another specific fluctuation by means of Sch Ödinger’s equation and the wave function |〉i ψ. Therefore, the fluctuations )(〉〈−i x x and )(〉〈−i x x p p must be the statistical configurationsof these fluctuations, namely, the fluctuations )(〉〈−i x x and )(〉〈−i x x p p must be thought of as the sum over these fluctuations caused by various reasons. Considering that a fluctuation caused by one of these reasons is independent ofanother, we can say the fluctuations )(〉〈−i x x and )(〉〈−i x x p p obeythe central limit theorem, and the fluctuations accord with Gaussian distribution []7,6:]})()()()([21exp{21),(2222i i i x x x i i x i x i p p p x x x p x p x f ∆〉〈−+∆〉〈−−∆∆=π (4) i x ∆ and i x p ∆ are given by Eq.(2), and they obey the uncertainty relation. Beingdifferent from the traditional fluctuations’ theory, taking quantum effects into account, the fluctuations can exist in the equilibrium state, which means that the fluctuations will not change the microscopic-states number so that the entropy of the system still is the greatest. We noticed ),(x i p x f and Einstein’s formula offluctuations have the same form of function, an important difference between them is that the variables x and x p of ),(x i p x f only correspond to the equilibrium state, the variables x and x p of Einstein’s formula only do to thenon-equilibrium state, which means while Einstein’s formula is valid the number of microscopic states will change to turn the system’s entropy into smaller. The difference between Eq.(4) and Einstein’s formula is determined by the value of i x i p x ∆∆ which affects the probability density of the fluctuation distribution.At first, we analyze the fluctuations qualitatively. Suppose that a system is in the equilibrium state, but the inequality of the uncertainty relation is established. It is clear that in the non-equilibrium state far away from the equilibrium state thefluctuations are very great which the inequality of the uncertainty relation satisfies, but the system do be in the non-equilibrium state. Even if in the neighborhood of the equilibrium state the entropy of system itself is a Lyapounov function to the isolated system, which makes the fluctuations decrease to the smallest and turns the system’s state into the equilibrium state []7. Thus, the case is impossible, and the equality of the uncertainty relation should be valid in the equilibrium state.When 〉〈=i x x , 〉〈=i x x p p , x i p x f ,() of Eq. (4) takes the formii x i x i i p x p x f ∆∆=〉〈〉〈π21),( (5) Since the fluctuations in the equilibrium state are the smallest, Eq.(5) should be the greatest, which guarantees the subsystem to be in a statistical average state, 〉〈=i x x and 〉〈=i x x p p , for the longest time to meet the requirements of theensemble theory. Obviously, only the minimal uncertainty relation can lead to this situation, which means the minimal uncertainty relation corresponds to theequilibrium state. This conclusion is in accord with the above qualitative analysis. The particles’ number of the equilibrium-state fluctuations is greater than the particles’ number of the non-equilibrium-state fluctuations in the same area nearby 〉〈i x and 〉〈i p , but the situation is converse in an area far away from 〉〈i x and 〉〈i x p . As 2)(i x ∆ and 2)(i x p ∆ are changeable, and what is their regular pattern?Using the minimal uncertainty relation: 22)]4/([)(i x x h p i ∆=∆π, while 〉〈≠〉〈≠i x x i p p x x , and x , x p are very close to 〉〈〉〈i x i p x , and aretemporally considered as constant for the change of 2)(i x ∆, let the first partialderivative of ),(x i p x f with respect to 2)(i x ∆ be zero, which indicates that ),(x i p x f still keeps an extreme value to make the fluctuations the smallest, we then have||4)(2〉〈−〉〈−=∆i x x i i p p x x h x π , ||4)(2〉〈−〉〈−=∆i x x x x x p p h p i i π(6) Since the second partial derivative of ),(x i p x f with respect to 2)(i x ∆ isnegative while Eq.(6) is valid, it has a maximum. With the same reason for the ensemble:]})()()()([21exp{21),(2222x x x x x p p p x x x p x p x F ∆〉〈−+∆〉〈−−∆∆=π (7) 222)4()()(πh p x x ≥∆∆ (8) ||4)(2〉〈−〉〈−=∆x x p p x x h x π , ||4)(2〉〈−〉〈−=∆x x p p h p x x x π(9) In the diagonal representations of the normalized ρ: 1)(=ρTr ,)(x Tr x ρ=〉〈 , )(x x p Tr p ρ=〉〈 (10)])([)(22〉〈−=∆x x Tr x ρ , ])([)(22〉〈−=∆x x x p p Tr p ρ (11) The probability density ),(x p x F conforms to the minimal uncertainty relation, describing the ensemble’s fluctuations. It is important that the number of microscopic states will not change while Eq.(7) is established, which means the probability i ρ of Eq.(1) will not change for all possible states, being distinct from the traditional ensemble-fluctuations theory in which the fluctuations will change the microscopic states of the ensemble and will turn the ensemble state into the non-equilibrium state. It is interesting that the changes of 2)(x ∆ and 2)(x p ∆ [or of 2)(i x ∆ and 2)(i x p ∆] are adjusted by the absolute value of ratio of )(〉〈−x x to )(〉〈−x x p p[or )(〉〈−i x x to )(〉〈−i x x p p ], obeying the minimal uncertainty relation, althoughthey are very average values. When the inequality of the uncertainty relation isestablished, namely, x p x ∆∆ and i x i p x ∆∆ become greater,),(x p x F and ),(x i p x f become smaller for the same 〉〈x , 〉〈x p and 〉〈i x , 〉〈i x p , whichmeans the fluctuations are amplified and the ensemble’s state and the subsystem’s state are turned into the non-equilibrium state. Obviously, Sch Ödinger’s equation and ρ cannot interpret these characters. So far, we proved the equality of the uncertainty relation corresponds to the equilibrium state (the entropy )0S , the inequality does to the non-equilibrium state (the entropy S ), Eq.(8) and the formula of the second law of thermodynamics, S S ≥0, are one-to-one, and the uncertainty relation is aquantum-mechanics expression of the law. Using the minimal uncertainty relation and substituting Eqs.(6) and (9) in Eqs.(4) and (7) respectively, we obtain brief expressions: |]))((|4exp[2),(〉〈−〉〈−−=i x x i x i p p x x hh p x f π (12a) |]))((|4exp[2),(〉〈−〉〈−−=x x x p p x x h h p x F π (12b) Equations (12a) and (12b) are to say, the fluctuations have curves of constant distributions although 2)(i x ∆, 2)(i x p ∆, 2)(x ∆ and 2)(x p all are changeable at the same time. When the minimal uncertainty relations in Eqs.(3) and (8) are valid, the minimal uncertainty relations of time-energy are hold []8,1: π4h t E i i =∆∆, π4h t E =∆∆ (13) 〉〉〈−〈=∆i i i i E E E ψψ|)(|)(2, ])([)(22〉〈−=∆E E Tr E ρ (14) 〉〈=〉〈i i i E E ψψ||, )(E Tr E ρ=〉〈 (15) Being different from x (or i x ), x p (or i x p ) and E (or i E ) which are the ensemble’s variables (or the subsystem’s variables), t and i t are belong to a reference system, which is out of the ensemble []8,1, and they act as reference variables. Since i E ∆ and E ∆ are the energy fluctuations of the subsystem and the energy fluctuations of the ensemble respectively in the equilibrium state, t ∆ is the interval of time while the ensemble’s energy undergoes fluctuations E ∆, and it is also the undergoing time of the equilibrium state. Because the subsystem is in the ensemble, i t ∆ should equal to t ∆, we furthermore obtainii t h t h E E ∆=∆=∆=∆ππ44 (16) Equation (16) indicates i E ∆ and E ∆ have self-similarity, which cannot beamplified; so do i x i p x ∆∆and x p x ∆∆. While there is a continuous phase transition, the self-similarity of correlation length is resulted from fluctuations, which are amplified to the infinite []7,6. Both kinds of fluctuations are two distinct limit situations.As the minimal uncertainty variables are the fluctuations of the equilibrium state, when a system’s state is measured by an apparatus, the system is disturbed and the system’s state is changed to the non-equilibrium state so that the minimal uncertainty variables are unmeasured, which are considered as limits of measured values. Finally, we suppose that because Eqs. (6) and (9) become invalid while 〉〈=i x x , 〉〈=i x x p p and 〉〈=x x , 〉〈=x x p p , and i x ∆, i x p ∆ and x ∆, x p ∆ will not equal to zero forever, they must keep the smallest simultaneously so that the sum )(i x i p x ∆+∆ and the sum )(x p x ∆+∆ must be the smallest restricted by the minimal uncertainty relation, which leads to results: 2/1)/(21πh p x i x i =∆=∆ while 〉〈=i x x and 〉〈=i x x p p ;2/1)/(21πh p x x =∆=∆ while 〉〈=x x and 〉〈=x x p p . The assumption is waiting for examination.[]1 William C. Price and Seymour S. Chissick, The Uncertainty Principle and Foundations of Quantum Mechanics (John Wiley & Sons, London, 1977) pp 7,13[]2 L. D. Landau and E. M. Lifshitz, Statistical Physics, Part.1, third-ed. (Butterworth-Heinemann, Oxford, 1980) pp 17,333[]3 Christian Maes and Karel Netocny, J. Statist. Phys, 110, 269-310 (2003)[]4 E. T. Jaynes, Ann.Rev.Phys.Chem.31, 579-600 (1980)[]5A. V . Golovnev and L.V .Prokhorov, J.Phys.A: Math.Gen. 37, 2765-2775 (2004)[]6Leo.P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalizations (World Science Publishing, London, 2000) pp 29,61[]7 L. E. Reichl, A Modern Course in Statistical Physics (The University of Texas Press, Austin, 1980) pp 147,510,545,689,318[]8L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon Press, Oxford, 1977) pp 157。

finite element method;Finite element method (FEM) is a numerical technique used to approximate the solutions of differential equations. It is widely used in the field of structural analysis, heat transfer, fluid dynamics, electromagnetics, and many other engineering and scientific disciplines.In FEM, a complicated geometry or physical domain is divided into smaller, simpler regions called finite elements. These elements are connected at specific points called nodes. By applying appropriate mathematical techniques and numerical algorithms, differential equations governing the behavior of the system under consideration are converted into a system of algebraic equations. This system of equations can then be solved using numerical methods to obtain an approximation of the desired solution.The finite element method provides several advantages over traditional analytical methods, particularly when dealing with complex geometries or nonlinear material behavior. Some key benefits include:1. Flexibility: FEM can handle various types of boundary conditions, material properties, and geometry configurations, making it highly flexible for modeling complex physical systems.2. Accuracy: By using a large number of finite elements, FEM can achieve high accuracy and precision in approximating the solutions of differential equations.3. Adaptability: The size and shape of finite elements can bemodified to capture specific phenomena or regions of interest more accurately.4. Versatility: FEM can handle a wide range of physical phenomena, including static and dynamic problems, linear and nonlinear behavior, and coupled systems.However, FEM also has certain limitations and challenges. It can be computationally expensive and time-consuming, especially for large-scale problems. Additionally, proper mesh generation and selection of appropriate element types and material models are crucial for obtaining accurate results.Despite these challenges, the finite element method remains a powerful and widely used numerical technique in engineering and scientific research. It has significantly contributed to the advancement of various fields by enabling the analysis and design of complex systems that would be difficult or impossible to solve analytically.。

anhydrous for analysis emsure -回复Anhydrous for Analysis EMSURE: Understanding Its Importance and ApplicationsIntroduction:Anhydrous for Analysis EMSURE is a high-quality reagent widely used in various scientific disciplines and industries. It plays a crucial role in ensuring accurate and reliable analytical results. In this article, we will explore in detail the significance, properties, and applications of Anhydrous for Analysis EMSURE, thereby providing a comprehensive understanding of this essential reagent.1. What is Anhydrous for Analysis EMSURE?Anhydrous for Analysis EMSURE is a term used to describe a broad range of reagents that are completely free from water molecules. These reagents are produced using advanced techniques to remove any moisture content, ensuring maximum stability and purity. Anhydrous for Analysis EMSURE is typically available in ultra-pure forms, meeting the highest quality standards demanded by analytical laboratories.2. Importance of Anhydrous for Analysis EMSURE:2.1. Eliminating Water Interference:Water is a common impurity in many chemicals used in analytical processes. However, the presence of water can interfere with various reactions and measurements, leading to inaccurate results. Anhydrous for Analysis EMSURE eliminates this interference, allowing for precise and reliable analysis.2.2. Enhanced Stability:Water can initiate degradation processes in certain substances, affecting their stability over time. Anhydrous for Analysis EMSURE, being entirely free from water, exhibits superior stability and prolonged shelf life. This property is especially critical for long-term storage of reagents and standards.2.3. Prevention of Hydrate Formation:Certain compounds readily react with water, forming hydrates—a chemically combined form where water molecules are incorporated into the substance's crystal lattice. Anhydrous for Analysis EMSURE prevents hydrate formation, maintaining the integrity of thecompound and ensuring accurate analysis.3. Properties of Anhydrous for Analysis EMSURE:3.1. Low Water Content:Anhydrous for Analysis EMSURE reagents typically have an extremely low moisture content, often in the range of parts per million (ppm) or below. This ensures minimal water-related interference during analytical procedures.3.2. High Purity:To meet the stringent requirements of analytical applications, Anhydrous for Analysis EMSURE reagents are manufactured to possess high purity levels. They undergo rigorous quality control measures, including multiple purification steps, to eliminate impurities that could affect the accuracy of analytical results.3.3. Traceable Certification:Anhydrous for Analysis EMSURE reagents are accompanied by comprehensive certificates of analysis, detailing the quality, purity, and conformity of the product. These certificates provide traceability and help maintain consistency in analytical procedures.4. Applications of Anhydrous for Analysis EMSURE:4.1. Chemical Analysis:Anhydrous for Analysis EMSURE reagents are widely used in various chemical analyses, including titrations, spectrophotometry, chromatography, and atomic absorption spectroscopy. Their water-free nature ensures accurate measurements and consistent results.4.2. Pharmaceutical Industry:In the pharmaceutical industry, Anhydrous for Analysis EMSURE is invaluable for conducting quality control tests, formulation development, and stability studies. It helps ensure the purity and stability of drug substances and excipients, thus contributing to the production of safe and effective medications.4.3. Food and Beverage Industry:Anhydrous for Analysis EMSURE reagents find extensive utility in the food and beverage industry. They are employed for the analysis of food components, additives, and contaminants, ensuring compliance with regulatory standards and ensuring consumersafety.4.4. Environmental Analysis:In environmental analysis, Anhydrous for Analysis EMSURE reagents aid in monitoring pollution levels, assessing the quality of water and air, and investigating the impact of pollutants on the environment. The absence of water interference allows for precise measurements and reliable data.5. Conclusion:Anhydrous for Analysis EMSURE is an indispensable reagent that plays a vital role in ensuring accurate and reliable analysis across various scientific disciplines and industries. Its ability to eliminate water interference, enhance stability, and prevent hydrate formation makes it a preferred choice for a wide range of applications. By understanding the significance and properties of Anhydrous for Analysis EMSURE, researchers and analysts can confidently employ this high-quality reagent to obtain precise and consistent results.。

The significance of froth stability in mineralflotation — A review中文翻译******班级：化工08-1班学号：**********The significance of froth stability in mineral flotation — A review矿物浮选中泡沫稳定性的意义-评论Abstract摘要This paper presents a review of the published articles related to froth stability and its importance in mineral flotation. Froth structure and froth stability are known to play a significant role in determining the mineral grade and recovery achieved in a flotation operation. Froth stability is depending not only on the type and concentration of the frother but also on the nature and amount of the particles present in the system. To date, there is no specific criterion to quantify froth stability although a number of parameters are used as indicators of froth stability. Linking froth stability to the metallurgical performance is also challenged.文章综述了已发表文章涉及矿物浮选过程中泡沫稳定性及其重要性的相关观点。